The helicopter in the drawing is moving horizontally to the right at a constant velocity. The weight of the helicopter is W=59900 N. The lift force L generated by the rotating blade makes an angle of 21.0° with respect to the vertical. (a) What is the magnitude of the lift force? (b) Determine the magnitude of the air resistance R that opposes the motion.
To solve this problem, we first need to determine the components of the lift force and the weight force.
(a) Magnitude of lift force (L):
To find the magnitude of the lift force, we can use trigonometry. The vertical component of the lift force can be calculated as:
L_vertical = L * sin(21.0°)
Given that the weight of the helicopter is W = 59900 N, the magnitude of the lift force is equal to the magnitude of the weight force:
L_vertical = W = 59900 N
(b) Magnitude of air resistance (R):
Since the helicopter is moving horizontally at a constant velocity, the air resistance force must be equal in magnitude and opposite in direction to the net horizontal force acting on it.
The net horizontal force acting on the helicopter is the component of the lift force perpendicular to the motion of the helicopter. This component can be represented as:
L_horizontal = L * cos(21.0°)
Since the lift force and air resistance force are equal in magnitude and opposite in direction, we can say that:
L_horizontal = R
Therefore, to find the magnitude of the air resistance (R), we need to calculate the horizontal component of the lift force:
R = L_horizontal = L * cos(21.0°)
Please provide the value of L (the magnitude of the lift force) to determine the magnitude of the air resistance (R) in Part (b).
To find the magnitude of the lift force, we can decompose it into vertical and horizontal components. Since the lift force makes an angle of 21.0° with respect to the vertical, we can use trigonometry to find the vertical component of the lift force.
(a) Magnitude of the lift force:
The vertical component of the lift force is given by L_vertical = L * sin(21.0°), where L is the magnitude of the lift force. We are given that the weight of the helicopter is W = 59900 N, which is equal to the magnitude of the vertical component of the lift force. Therefore, we can write:
W = L * sin(21.0°)
Solving for L, we get:
L = W / sin(21.0°)
Now we can substitute the given values to find the magnitude of the lift force:
L = 59900 N / sin(21.0°) ≈ 135,730.14 N
Therefore, the magnitude of the lift force is approximately 135,730.14 N.
(b) To determine the magnitude of the air resistance that opposes the motion, we need to consider that the helicopter is moving horizontally to the right at a constant velocity. This means that the net force acting on the helicopter is zero.
The net force acting on the helicopter in the horizontal direction is the difference between the air resistance (R) and the horizontal component of the lift force (L_horizontal). Since the helicopter is moving at a constant velocity, the lift force must balance out the air resistance. Therefore, we can write:
R = L_horizontal
The horizontal component of the lift force can be found using trigonometry:
L_horizontal = L * cos(21.0°)
Substituting the given values, we get:
L_horizontal = 135,730.14 N * cos(21.0°) ≈ 125,227.97 N
Therefore, the magnitude of the air resistance that opposes the motion is approximately 125,227.97 N.